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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaiun1 | Structured version Visualization version GIF version |
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.) |
Ref | Expression |
---|---|
imaiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3285 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3481 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 6086 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
4 | 3 | rexbii 3091 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
5 | eliun 4999 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) | |
6 | 5 | anbi2i 623 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) |
7 | r19.42v 3188 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
8 | 6, 7 | bitr4i 278 | . . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1844 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
10 | 1, 4, 9 | 3bitr4ri 304 | . . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) |
11 | 2 | elima3 6086 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
12 | eliun 4999 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) | |
13 | 10, 11, 12 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)) |
14 | 13 | eqriv 2731 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∃wrex 3067 〈cop 4636 ∪ ciun 4995 “ cima 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-iun 4997 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 |
This theorem is referenced by: trclimalb2 43715 |
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